You should not use SQLite in production.
I've deployed SQLite to a large number of production software, from embedded systems to web servers handling thousands of daily vistors. In fact, SQLite is my tool of choice when it comes to a small/medium relational-db-based web project.
Sqlite is not a toy db. It is an impressive piece of software engineering.
So, unless you have a very narrow definition of the word production, you should do read up on SQLite (the official website is excellent) and seriously consider it for your next project.